Fig. 4

Different types of stochastic QSSA. a The diagram of full model (Eqs. 11-14) and two types of reduced models (Eqs. 17 and 18). b When an initial condition is on the limit cycle, only the stochastic tQSSA is accurate while both deterministic pQSSA and tQSSA are accurate (Additional file 1: Figure S4A). A detailed numerical error analysis with Fourier transform is provided in Additional file 1: Figure S5. c Even with an initial condition on the limit cycle (black circle), for which deterministic pQSSA is accurate (Additional file 1: Figure S4A), the stochastic trajectory escapes the limit cycle due to the fluctuation, where the deterministic pQSSA is inaccurate (e.g. black square) (Additional file 1: Figure S4B). The parameters of the model are adopted from [30]: α _M =15.1745h ⁻¹, α _P =α _F =β _M =β _P =β _F =1h ⁻¹, k _b =50h ⁻¹, k _f =200n M ⁻¹ h ⁻¹, D _T =164.75n M